Previous |  Up |  Next

Article

Keywords:
Hermitian symmetric spaces; standard operators; BGG sequence; Hasse diagram; weight graph
Summary:
This paper gives a description of a method of direct construction of the BGG sequences of invariant operators on manifolds with AHS structures on the base of representation theoretical data of the Lie algebra defining the AHS structure. Several examples of the method are shown.
References:
[1] Baston R.J.: Almost Hermitian symmetric manifolds, I: Local twistor theory, II: Differential invariants. Duke Math. J. 63 (1991), 81-111, 113-138. MR 1106939
[2] Baston R.J., Eastwood M.G.: Penrose Transform; Its Interaction with Representation Theory. Clarendon Press, Oxford, 1989. MR 1038279 | Zbl 0726.58004
[3] Bernstein I.N., Gelfand I.M., Gelfand S.I.: Structure of representations generated vectors of highest weight. Funct. Anal. Appl. 5 (1971), 1-8. MR 0291204
[4] Bernstein I.N., Gelfand I.M., Gelfand S.I.: Differential operators on the base affine space and a study of $\frak g$-modules. in ``Lie Groups and their Representations'' (ed. I.M. Gelfand) Adam Hilger, 1975, pp.21-64. MR 0578996
[5] Branson T., Ólafsson G., Ørsted B.: Spectrum generating operators and intertwining operators for representations induced from a maximal parabolic subgroup. J. Funct. Anal. 135 (1996), 163-205. MR 1367629
[6] Bureš J.: Special invariant operators I. Comment. Math. Univ. Carolinae 37.1 (1996), 179-198. MR 1396170
[7] Čap A.: Translation of natural operators on manifolds with AHS-structures. Archivum Math. (Brno) 32.4 (1996), 249-266, electronically available at www.emis.de. MR 1441397
[8] Čap A., Schichl H.: Parabolic Geometries and Canonical Cartan Connections. preprint ESI 450, electronically available at www.esi.ac.at. MR 1795487
[9] Čap A., Slovák J., Souček V.: Invariant operators on manifolds with almost Hermitian symmetric structures, I. Invariant differentiation. Acta Math. Univ. Commenianae 66 (1997), 33-69, electronically available at www.emis.de. MR 1474550
[10] Čap A., Slovák J., Souček V.: Invariant operators on manifolds with almost Hermitian symmetric structures, II. Normal Cartan connections, Acta Math. Univ. Commenianae. 66 (1997), 203-220, electronically available at www.emis.de. MR 1620484
[11] Čap, A., Slovák J., Souček V.: Invariant operators on manifolds with almost Hermitian symmetric structures, III. Standard Operators. ESI Preprint 613, to appear in J. Differential Geom. Appl., electronically available at www.esi.ac.at. MR 1757020 | Zbl 0969.53004
[12] Eastwood M.G.: On the weights of conformally invariant operators. Twistor Newsl. 24 (1987), 20-23.
[13] Eastwood M.G., Slovák J.: Semi-holonomic Verma modules. J. Algebra 197 (1997), 424-448. MR 1483772
[14] Fegan H.D.: Conformally invariant first order differential operators. Quart. J. Math. 27 (1976), 371-378. MR 0482879 | Zbl 0334.58016
[15] Fulton W., Harris J.: Representation Theory - A First Course. Springer-Verlag (GTM), 1991. MR 1153249 | Zbl 0744.22001
[16] Garland H., Lepowsky J.: Lie Algebra Homology and the Macdonald-Kac Formulae. Inv. Math. 34, Springer, 1976. MR 0414645
[17] Gindikin S.G.: Generalized conformal structures. Twistors in Mathematics and Physics, LMS Lecture Notes 156, Cambridge Univ. Press, 1990, pp.36-52. MR 1089908 | Zbl 0788.22008
[18] Goncharov A.B.: Generalized conformal structures on manifolds. Selecta Math. Soviet. 6 (1987), 308-340. MR 0925263 | Zbl 0632.53038
[19] Humphreys J.E.: Introduction to Lie Algebras and Representation Theory. Springer-Verlag, 1972. MR 0323842 | Zbl 0447.17002
[20] Jakobsen H.P.: Conformal invariants. Publ. RIMS, Kyoto Univ. 22 (1986), 345-361. MR 0849262
[21] Jacobson N.: Lie Algebras. Interscience Tracts, No. 10, 1962. MR 0143793 | Zbl 0333.17009
[22] Johnson K.D.: Decomposition of Exterior Algebras. Contemp. Math. 191, AMS, 1995. MR 1365537 | Zbl 0851.57040
[23] Kobayashi S., Nagano T.: On filtered Lie algebras and geometric structures I. J. Math. Mech. 13 (1964), 875-907. MR 0168704 | Zbl 0142.19504
[24] Kolář I., Michor P.W., Slovák J.: Natural Operations in Differential Geometry. Springer, 1993. MR 1202431
[25] Lepowsky J.: A generalization of the Bernstein-Gelfand-Gelfand resolution. J. Algebra 49 (1977), 496-511. MR 0476813 | Zbl 0381.17006
[26] Sharpe R.W.: Differential Geometry. Graduate Texts in Mathematics 166, Springer-Verlag, 1997. MR 1453120 | Zbl 0876.53001
[27] Slovák J.: On the geometry of almost Hermitian symmetric structures. in Proceedings of the Conference Differential Geometry and Applications, 1995, Brno, Masaryk University, Brno (1996), pp.191-206, electronically available at www.emis.de. MR 1406338
[28] Slovák J.: Parabolic geometries. Research Lecture Notes, Part of DrSc. Dissertation, Preprint IGA 11/97, electronically available at www.maths.adelaide.edu.au.
[29] Verma D.N.: Structure of certain induced representations of complex semisimple Lie algebras. Bull. Amer. Math. Soc. 74 (1968), 160-166. MR 0218417 | Zbl 0157.07604
[30] Wünsch V.: On conformally invariant differential operators. Math. Nachr. 129 (1986), 269-281. MR 0864639
Partner of
EuDML logo